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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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§10. Parabolic Fixed Points: The Leau-Fatou Flower

Again we consider functions f(z) = λz + a2z2 + a3z3 +... which are defined and holomorphic in some neighborhood of the origin, but in this section we suppose that the multiplier λ at the fixed point is a root of unity, λq = 1. Such a fixed point is said to be parabolic, provided that fimageq is not the identity map. (Compare Lemma 4.7.) First consider the special case λ = 1. Then we can write our map as

image

with n ≥ 1 and a ≠ 0. The integer n + 1 is called the multiplicity of the fixed point. (Compare Lemma ...

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